1. Triangular coordinates and special element
Construct the shape functions N2 and N5 for the special triangular element sketched on the right. The element is expected to be used to connect two quadratic elements into one linear element. The interpolation should be linear along the between nodes 1 and 2, and quadratic along the other two edges. Write your results in terms of the triangular coordinates, and sketch the shapes of the two shape functions. Make sure that your shape functions satisfy the requirements.
2. Rectangular element Given a nine-node rectangular element as shown on the right
1) Construct the element shape functions by the tensor product method. (You may just write them in terms of the 1D shape functions. No need to substitute or expand.)
2) Suppose you are interpolating a temperature field θ(x, y) with the shape functions. If the temperature at nodes A and B is 100°C and 0°C at all other nodes. What is the interpolated temperature on point (1,1)? Feel free to use matlab or other software for the calculation.
3) Consider the 6-node triangular element located to the right of the rectangular element. Will the interpolated function be continuous across the edge AB? Explain.
3. Jacobian of elements
Calculate the Jacobian (determinant of the Jacobian matrix) J(ξ, η) for 4-node quadrilateral elements (isoparametric) with the following sets of nodes. Plot the distribution of ?? on the (ξ, η) plane (or (x, y) plane if you can). Use whatever software you feel comfortable with for the calculation and plotting. Discuss the results.
The coordinates of the 4 nodes in real space (the sequence is the numbering sequence):
1) (0,-1), (1, 0), (0, 1), (-1, 0)
2) (-3,0), (2, 0), (2, 2), (-2, 2)
3) (0, 0), (-1, -1), (2, 0), (-1, 1)